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Similar Triangles in Geometry

Similar triangles are two triangles that have the same shape, but not necessarily the same size. They have the same angles and corresponding sides that are in proportion to one another. Geometric similarity is when two or more geometric figures have the same shape, but not necessarily the same size. Similar triangles are a special case of geometric similarity.

Similarity Theorem

The similarity theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. This means that if two angles of one triangle are congruent to two angles of another triangle, the sides of the two triangles must be in proportion to each other. The similarity theorem is also known as the AAA theorem.

Congruence Theorem

The congruence theorem states that if two triangles have all three sides in proportion to each other, then the two triangles are similar. This means that if two triangles have corresponding sides that are in proportion to each other, then the angles of the two triangles must be congruent. The congruence theorem is also known as the SSS theorem.

Triangle Congruence

Triangle congruence is when two triangles have the same size and shape. This means that all three sides of the two triangles must be equal in length, and all three angles of the two triangles must be equal in measure. The two triangles are also referred to as being congruent. The congruence theorem is also known as the SAS theorem.

Practice Problems

1. In triangle ABC, the measures of angles A and C are both 60. In triangle DEF, the measures of angles D and F are both 45. Are the two triangles similar?

Answer: No, the two triangles are not similar because the measures of the angles are not congruent. The angles of triangle ABC are both 60, and the angles of triangle DEF are both 45. Since the angles are not congruent, the two triangles are not similar.

2. In triangle ABC, the length of side AB is 6 cm, and the length of side AC is 10 cm. In triangle DEF, the length of side DE is 8 cm, and the length of side EF is 12 cm. Are the two triangles similar?

Answer: Yes, the two triangles are similar because the corresponding sides are in proportion to each other. The length of side AB is 6 cm, and the length of side DE is 8 cm. The length of side AC is 10 cm, and the length of side EF is 12 cm. Since the sides are in proportion to each other, the two triangles are similar.

3. In triangle ABC, the length of side AB is 3 cm, and the length of side BC is 4 cm. In triangle ABC, the length of side AC is 5 cm. Are the two triangles similar?

Answer: Yes, the two triangles are similar because the sides of the two triangles are in proportion to each other. The length of side AB is 3 cm, and the length of side AC is 5 cm. The length of side BC is 4 cm, and the length of side AC is 5 cm. Since the sides are in proportion to each other, the two triangles are similar.

Summary

Similar triangles are two triangles that have the same shape, but not necessarily the same size. They have the same angles and corresponding sides that are in proportion to each other. The similarity theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. The congruence theorem states that if two triangles have all three sides in proportion to each other, then the two triangles are similar. Triangle congruence is when two triangles have the same size and shape. The congruence theorem is also known as the SAS theorem.

FAQ

What is the definition of similar triangles?

Similar triangles are two triangles with the same shape but different sizes. They have all their corresponding angles equal and all corresponding sides in the same proportion.

What is the congruence theorem for similar triangles?

The congruence theorem for similar triangles states that if two triangles are similar, then the corresponding sides of the triangles have the same ratio. This means that the sides of the two triangles have the same length, or that the ratio of the lengths of any two sides of one triangle is equal to the ratio of the lengths of the corresponding sides of the other triangle.

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